Difference between revisions of "2001 IMO Shortlist Problems/N6"
(New page: == Problem == Is it possible to find 100 positive integers not exceeding 25,000, such that all pairwise sums of them are different? == Solution == {{solution}} == Resources == * [[2001 ...) |
(This should have been combinatorics, and how did it get to the shortlist anyways???) |
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== Solution == | == Solution == | ||
| − | {{ | + | The biggest pairwise sum is <math>24999+25000=49999</math>, and there are <math>\binom{100}{2}=50050</math> sums. Thus by the [[Pigeonhole Principle]], there must be at least two sums which are equal. |
== Resources == | == Resources == | ||
Revision as of 09:12, 6 October 2008
Problem
Is it possible to find 100 positive integers not exceeding 25,000, such that all pairwise sums of them are different?
Solution
The biggest pairwise sum is 
, and there are 
sums. Thus by the Pigeonhole Principle, there must be at least two sums which are equal.
